I am studying for my 2nd midterm, but then I am stuck..plz help me..thank you so much.

1) prove by induction that for each nonnegative integer k, the integer k^2 + 5k is even.

2) The number n!, which is the number of permutations of the first n positive integers, is defined recursively by 0! = 1 and n! = n X (n-1)! for all n > o. Prove that n! < n ^n for all n > 0 and strict inequality holds if n > 1.

Nov 14th 2006, 07:46 PM

ThePerfectHacker

Quote:

Originally Posted by jenjen

1) prove by induction that for each nonnegative integer k, the integer k^2 + 5k is even.

I let somebody else do this problem. It is just boring.

Quote:

2) The number n!, which is the number of permutations of the first n positive integers, is defined recursively by 0! = 1 and n! = n X (n-1)! for all n > o. Prove that n! < n ^n for all n > 0 and strict inequality holds if n > 1.

We see that,
Good.

Next there is a such as,

But,
Multiply by both sides by
Thus,
Q.E.D.

Nov 14th 2006, 07:49 PM

topsquark

Quote:

Originally Posted by jenjen

1) prove by induction that for each nonnegative integer k, the integer k^2 + 5k is even.

Let k = 0. Then is even.

Let the theorem be true for some k = n.

Then we need to show that the theorem is true for k = n + 1. That is: is even.

So

By hypothesis is even. The second term 2(n + 3) is a number multiplied by 2, which is even by definition. Thus the problem is in the form of an even number plus an even number, which is always even.

-Dan

Nov 14th 2006, 08:25 PM

jenjen

thank you so much ThePerfectHacker and Topsquark for the quick replies!!!!!